Boolean Matrix Multiplication for Highly Clustered Data on the Congested Clique

TL;DR

This paper introduces an efficient protocol for Boolean matrix multiplication tailored for highly clustered data in the congested clique model, leveraging approximate MST computations in Hamming space to optimize round complexity.

Contribution

It presents a novel protocol for Boolean matrix multiplication that exploits data clustering and provides an efficient approximate MST algorithm in Hamming space.

Findings

Uses (\u221A(M/n)+1) rounds for matrix multiplication with high probability.

Provides a protocol for approximate MST in Hamming space with O((\,log^3 n)) rounds.

Achieves efficient Boolean matrix multiplication for highly clustered data in distributed settings.

Abstract

We present a protocol for the Boolean matrix product of two $n\times b$ Boolean matrices on the congested clique designed for the situation when the rows of the first matrix or the columns of the second matrix are highly clustered in the space $\{0,1\}^n.$ With high probability (w.h.p), it uses $\tilde{O}\left(\sqrt {\frac M n+1}\right)$ rounds on the congested clique with $n$ nodes, where $M$ is the minimum of the cost of a minimum spanning tree (MST) of the rows of the first input matrix and the cost of an MST of the columns of the second input matrix in the Hamming space $\{0,1\}^n.$ A key step in our protocol is the computation of an approximate minimum spanning tree of a set of $n$ points in the space $\{0,1\}^n$. We provide a protocol for this problem (of interest in its own rights) based on a known randomized technique of dimension reduction in Hamming spaces. W.h.p., it constructs an $O(1)$-factor approximation of an MST of $n$ points in the Hamming space $\{ 0,\ 1\}^n$ using $O(\log^3 n)$ rounds on the congested clique with $n$ nodes.